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\begin{document}
\title{论文排版}
\author{王劼}
\section{不可压缩的Navier-Stokes方程}
  不可压缩流体的二维流场完全由速度矢量$q = (u(x, y), v(x, y)) \in \mathrm{R}^2$描述。\cite{hw2}\\
  这些函数是以下守恒定律的解(例如,参见赫希,1988)\\
  • 质量守恒：\\
  \begin{equation}\label{eq:11}
    \text{div}(q)=0
  \end{equation}\\
      或者，使用背离运算符\footnote{
      我们回顾微分算子散度、梯度和二维域的拉普拉斯式:如果$v = (v_x, v_y) : \mathrm{R}^2 \rightarrow \mathrm{R}^2
      并且\phi : \mathrm{R}^2 \rightarrow \mathrm{R},\\
      那么\text{div}(v) = \frac{{\partial v_x}}{{\partial x}} + \frac{{\partial v_y}}{{\partial y}}
      G\phi = \left( \frac{{\partial \phi}}{{\partial x}}, \frac{{\partial \phi}}{{\partial y}} \right)
      \Delta\phi = \text{div}(G\phi) = \frac{{\partial^2 \phi}}{{\partial x^2}} + \frac{{\partial^2 \phi}}{{\partial y^2}}
      \Delta v = (\Delta v_x, \Delta v_y)$
      }
    的显式形式编写，\\
    \begin{equation}\label{eq:12}
      \frac{{\partial u}}{{\partial x}} + 
      \frac{{\partial v}}{{\partial y}} = 0
    \end{equation}
  • 紧凑形式的动量守恒方程\footnote{
    我们用$\otimes$来表示张量积。
    }\\
    \begin{equation}\label{eq:13}
      \frac{{\partial q}}{{\partial t}} + \text{div}(q \otimes q) = -\mathcal{G} p + \frac{1}{Re} \Delta q
    \end{equation}
      或者，分离形式,\\
      \begin{equation}\label{eq:14}
      \begin{cases}
      \begin{aligned}   
      &\frac{\partial u}{\partial t} + \frac{\partial u^2}{\partial x} + \frac{\partial u v}{\partial y} = -\frac{\partial p}{\partial x} + \frac{1}{Re}\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right),\\
      &\frac{\partial v}{\partial t} + \frac{\partial u v}{\partial x} + \frac{\partial v^2}{\partial y} = -\frac{\partial p}{\partial y} + \frac{1}{Re}\left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}\right).
      \end{aligned}
      \end{cases}
    \end{equation}
  前面的方程以无量纲形式编写，使用以下缩放变量:\\
  \begin{equation}\label{eq:15}
    x = \frac{x^*}{L}, \quad y = \frac{y^*}{L}, \quad u = \frac{u^*}{V_0}, \quad v = \frac{v^*}{V_0}, \quad t = \frac{t^* L}{V_0}, \quad p = \frac{p^*}{\rho_0 V_0^2}
\end{equation}
  

其中上标(*)表示以物理单位测量的变量。这常数 $L$、$V_0$分别是表征模拟流动的参考长度和速度。\\
无量纲数 $Re$称为雷诺数数量并量化惯性(或对流)项的相对重要性和流动中的粘性(或扩散)项：\\
\begin{equation} \label{eq:16}
  Re = \frac{V_0L}{\nu} 
\end{equation}\\
其中$\nu$是流动的运动粘度。\\
总而言之，偏微分方程的纳维-斯托克斯系统将在数值上本项目中解决的问题由 \ref{eq:12} 和 \ref{eq:14} 定义;初始条件(在t = 0),\\
边界条件将在以下各节中讨论。\\

\section{计算域，交错网格，和边界条件}
数值求解Navier-Stokes方程是相当简化的的矩形域$L_x * L_y$(见图12.1)\\
到处都是边界条件。速度$q(x, y)$的周期性压力$p(x, y)$场在数学上表示为\\
\begin{equation}\label{eq:21}
  \forall y \in [0, L_y]: \quad q(0, y) = q(L_x, y), \quad p(0, y) = p(L_x, y),
\end{equation}\\
\begin{equation}\label{eq:22}
  \forall x \in [0, L_x]: \quad q(x, 0) = q(x, L_y), \quad p(x, 0) = p(x, L_y).
\end{equation}\\
  计算解的点分布在区域遵循矩形和均匀的二维网格。由于在我们的方法中并非所有的变量表都共享相同的网格，\\
因此我们首先定义一个主网格(参见图12.1)沿$x$方向取$n_x$个计算点,沿$y$方向取$n_y$个计算点生成:\\
\begin{equation}\label{eq:23}
  x_c(i) = (i - 1) \cdot \delta x, \delta x = \frac{L_x}{n_x - 1} ,i=1,...,n_x,
\end{equation}\\
\begin{equation}\label{eq:24}
  y_c(j) = (j - 1) \cdot \delta y, \delta y = \frac{L_y}{n_y - 1} ,j=1,...,n_y.
\end{equation}\\
\begin{tikzpicture}
  % 绘制坐标轴
  \draw[->] (-0.5,0) -- (2.5,0) node[right] {$L_x$};
  \draw[->] (0,-0.5) -- (0,2.5) node[above] {$L_y$};
  
  % 绘制正方形
  \draw[red] (0.1, 0) -- (0.1, 2.5) -- (2.6, 2.5) -- (2.6, 0) -- cycle;
  
  \draw[thick](-1,-1) -- (-1,4) -- (5,4) -- (5,-1) -- (-1,-1);
  \draw[<->] (0,1.25) -- (0.6,1.25) node[above] {periodicity};
  \draw[<->] (2.0,1.25) -- (2.9,1.25) node[above] {periodicity};
  \draw[<->] (1.3,1.5) -- (1.3,2.4) node[right] {periodicity};
  \draw[<->] (1.3,1.0) -- (1.3,0.1) node[right] {periodicity};
  \node[below right] at (1.1, 0) {$X$};
  \node[below left] at (0.1,1.45) {$Y$};
  \draw[thick](4.2,-1) -- (4.2,4) -- (11.2,4) -- (11.2,-1) -- (4.2,-1);
  \draw[thick] (7,3.5) -- (7,0.5) -- (10.8,0.5);
  \node[above right] at (7,3.3) {$L_y$};
  \node[above] at (10.5,0.5) {$L_x$};
  \node[below] at (10.7,0.5) {$X$};
  \node[rotate=90] at (6.8, 3.4) {$Y$};
  \draw[green](7,3.4) -- (10.4,3.4) -- (10.4,0.5);
  \draw[green](8.03,3.4) -- (8.03,0.5);
  \draw[green](9.06,3.4) -- (9.06,0.5);
  \draw[green](7,1.5) -- (10.4,1.5);
  \draw[green](7,2.45) -- (10.4,2.45);
  \draw[dashed](7,1.95) -- (10.4,1.95);
  \draw[dashed](8.7,3.4) -- (8.7,0.5);
  \node[above right] at (8.7,1.95) {$p(i,j)$};
  \node[above left] at (8.03,1.95) {$u(i,j)$};
  \node[below right] at (8.7,1.5) {$v(i,j)$};
  \node[left] at (7,2.45) {$y_c(j+1)$};
  \node[left] at (7,1.95) {$y_m(j)$};
  \node[left] at (7,1.5) {$y_c(j)$};
  \node[rotate=90] at (8.03,0) {$x_c(i)$};
  \node[rotate=90] at (8.7,0) {$x_m(i)$};
  \node[rotate=90] at (9.06,-0.2) {$x_c(i+1)$};
  \end{tikzpicture}\\
  Fig. $12.1.$ 计算域，交错网格和边界条件。\\
  次级网格由初级网格单元的中心定义:\\
  \begin{equation}\label{eq:25}
    x_m(i) = (i - \frac{1}{2}) \cdot \delta x, i=1,...,n_{xm},
  \end{equation}\\
  \begin{equation}\label{eq:26}
    y_m(j) = (j - \frac{1}{2}) \cdot \delta y, j=1,...,n_{ym}.
  \end{equation}\\
  其中我们使用了速记符号$n_{xm} = n_x-1,n_{ym} = n_y-1$。内部定义为矩形$[x_c(i), x_c(i+1)] \times [y_c(j), y_c(j+1)]$的计算单元,
  未知变量$u, v, p$将被计算为近似的不同空间位置的解决方案:\\
  \begin{itemize}
    \item $u(i, j)≈u(x_c(i), y_m(j))$(cell的西面),\\
    \item $v(i, j)≈v(x_m(i), y_c(j))$(胞体南侧),\\
    \item $p(i, j)≈p(x_m(i), y_m(j))$(细胞中心)。\\
  \end{itemize}


这种交错排列的变量有很强的优点压力和速度之间的耦合。它也有帮助(参见参考文献本章结束),\\
以避免一些稳定性和收敛的问题，如经验的并配安排(其中所有的变量计算在相同的网格点)。\\

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